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AE Version 2.0 11/09/18.
Always, sometimes, never?
AE Version 2.0 11/09/18.
Work and
energy
Avril Steele
Toby Rome
Simon Clay
AE Version 2.0 11/09/18.
Work & energy
AE Version 2.0 11/09/18.
Topics using energy Work, energy & power
• Definitions, work-energy principle,
conservation of energy
Elastic springs & strings
• Hooke’s law & elastic potential energy
Circular motion
• Vertical circular motion
Momentum & restitution
• Loss of kinetic energy in collisions
AE Version 2.0 11/09/18.
Specifications referencesAQA Edexcel OCR (A) OCR(B)
MEI
Work, energy
& power AS level FM1 AS AS level Mech a
Elastic springs
& stringsAS level FM1 A level Mech b
Circular
motionA level FM2 AS level Mech b
Momentum &
restitutionAS level FM1 AS AS level Mech a
AE Version 2.0 11/09/18.
Definitions
Kinetic Energy KE =
Gravitational Potential
EnergyGPE =
Elastic Potential Energy EPE =
Work Done WD =
𝐹 = force
𝑚 = mass
𝑣 = speed
𝑠 = distance moved
ℎ = height
𝜆 = modulus of elasticity
𝑥 = extension
𝑙 = natural length
𝑔 = acceleration due to gravity
AE Version 2.0 11/09/18.
Definitions
Kinetic Energy 𝐾𝐸 =1
2𝑚𝑣2
Gravitational Potential
Energy𝐺𝑃𝐸 = 𝑚𝑔ℎ
Elastic Potential Energy 𝐸𝑃𝐸 =𝜆𝑥2
2𝑙
Work Done 𝑊𝐷 = 𝐹𝑠
𝐹 = force
𝑚 = mass
𝑣 = speed
𝑠 = distance moved
ℎ = height
𝜆 = modulus of elasticity
𝑥 = extension
𝑙 = natural length
𝑔 = acceleration due to gravity
AE Version 2.0 11/09/18.
The work-energy principleHow do we get from Newton’s second law to the
work-energy principle… …and to conservation of
energy?
𝐹 = 𝑚 × 𝑎
Could be
variableLet’s keep
this constant
This is a rate
of change
AE Version 2.0 11/09/18.
The work-energy principle
𝐹 = 𝑚𝑎 = 𝑚d𝑣
d𝑡= 𝑚
d𝑥
d𝑡
d𝑣
d𝑥= 𝑚𝑣
d𝑣
d𝑥
𝐹 d𝑥 = 𝑚𝑣 d𝑣
Work done =1
2𝑚𝑣2
𝑣1
𝑣2
Work done =1
2𝑚𝑣2
2 −1
2𝑚𝑣1
2
𝑥 is the
displacement
Work done = Change in KE
AE Version 2.0 11/09/18.
Work doneBy a constant force
𝐹 moving a particle
a distance 𝑠 in the
direction of the
force
0
𝑠
𝐹 d𝑥 = 𝐹 𝑥 0𝑠 = 𝐹𝑠
By the weight of an
object as it moves
from height ℎ1 to
height ℎ2 (relative
to some datum)
ℎ1
ℎ2
−𝑚𝑔 d𝑥 = −𝑚𝑔 𝑥 ℎ1ℎ2 = −(𝑚𝑔ℎ2 −𝑚𝑔ℎ1)
By the tension
in a spring
stretching from
an extension 𝑥1to an extension 𝑥2
𝑥1
𝑥2
−𝜆𝑥
𝑙d𝑥 = −
𝜆
𝑙
1
2𝑥2
𝑥1
𝑥2
= −𝜆𝑥22
2𝑙−𝜆𝑥12
2𝑙
𝐹𝑠
𝑚𝑔ℎ1
ℎ2
𝑙 + 𝑥1
𝑙 + 𝑥2
AE Version 2.0 11/09/18.
The work-energy principle
𝐾𝐸 =1
2𝑚𝑣2 𝐺𝑃𝐸 = 𝑚𝑔ℎ 𝐸𝑃𝐸 =
𝜆𝑥2
2𝑙
Work done = change in KE + change in GPE + change in EPE
by ‘other’ forces
by all forces
Work done = Change in KE
𝐹𝑠 − (𝑚𝑔ℎ2 −𝑚𝑔ℎ1)−𝜆𝑥22
2𝑙−𝜆𝑥12
2𝑙=1
2𝑚𝑣2
2 −1
2𝑚𝑣1
2
AE Version 2.0 11/09/18.
The work-energy principle
𝐹𝑠 =1
2𝑚𝑣2
2 −1
2𝑚𝑣1
2 + (𝑚𝑔ℎ2 −𝑚𝑔ℎ1)+𝜆𝑥22
2𝑙−𝜆𝑥12
2𝑙
𝐾𝐸 =1
2𝑚𝑣2 𝐺𝑃𝐸 = 𝑚𝑔ℎ 𝐸𝑃𝐸 =
𝜆𝑥2
2𝑙
Work done = change in KE + change in GPE + change in EPE
by ‘other’ forces
by all forces
Work done = Change in KE
AE Version 2.0 11/09/18.
Conservation of energy
If there are no ‘other’ forces doing work then the total
change in energy is zero. Energy is conserved.
Work done = change in KE + change in GPE + change in EPE
AE Version 2.0 11/09/18.
AE Version 2.0 11/09/18.
Dropping a ball
What
assumptions are
we making?
AE Version 2.0 11/09/18.
Bungee jumperWhere should the release point be so that the tape
just touches the floor?
For red elastic
take 𝜆 = 6.5 N?
AE Version 2.0 11/09/18.
Calculations
Conservation of energy:
𝜆𝑥2
2𝑙= 𝑚𝑔(𝑙 + 𝑥)
𝜆𝑥2
2𝑙−𝑚𝑔𝑥 −𝑚𝑔𝑙 = 0
𝑙 + 𝑥
𝐾𝐸 = 0, 𝐺𝑃𝐸 = 𝑚𝑔 𝑙 + 𝑥 , 𝐸𝑃𝐸 = 0
𝐾𝐸 = 0, 𝐺𝑃𝐸 = 0, 𝐸𝑃𝐸 =𝜆𝑥2
2𝑙
release point
lowest point
The total distance fallen is 𝑙 + 𝑥, and
so the mass should be released from
rest at this height above the surface,
so that it just contacts the surface as it
comes to rest.
Don’t forgot to account for the size of
the object though!
AE Version 2.0 11/09/18.
Circular loops What’s the relationship between OA and OB for
the bob to complete a full circle about B?
AE Version 2.0 11/09/18.
Energy in circular motionCondition for particle on string to make full circle is
that 𝑢 > 5𝑔𝑟 at A.
At A: 𝐺𝑃𝐸 = 0, 𝐾𝐸 =1
2𝑚𝑢2
At B: 𝐺𝑃𝐸 = 2𝑚𝑔𝑟, 𝐾𝐸 =1
2𝑚𝑣2
O
A 𝑢 ms-1
𝑟 m
B
See handout for more
information on circular
motion.
AE Version 2.0 11/09/18.
Calculations
𝐾𝐸 = 0, 𝐺𝑃𝐸 = 𝑚𝑔𝑙
𝐾𝐸 =1
2𝑚(5𝑔𝑟), 𝐺𝑃𝐸 = 0
Conservation of energy:
5
2𝑚𝑔𝑟 = 𝑚𝑔𝑙
⇒ 𝑟 =2
5𝑙
The radius of the large arc is 𝑙.The radius of the small circle is 𝑟.
𝑂𝐵 = 𝑙 − 𝑟 ⇒ 𝑂𝐵 =3
5𝑂𝐴
𝑟
AE Version 2.0 11/09/18.
Where will it land?Is this statement justified?
Work done by friction = change in KE + change in GPE
AE Version 2.0 11/09/18.
Mechanics in actionElasticity
• Dangerous sports club (sheet 22)
Circular motion
• Looping-the-loop 1 & 2 (sheets 27 & 37)
• Wind up (sheet 38)
Collisions
• Bouncing balls 1 & 2 (sheets 49 & 50)
stem.org.uk/resources/elibrary/resource/26065/mechanics-action
2019 FM Conference
V1.1
Version 1.0
1 of 1
Work and energy
Specifications
AQA Edexcel OCR (A) OCR(B) MEI
Work, energy & power AS level FM1 AS AS level Mech a
Elastic springs & strings AS level FM1 A level Mech b
Circular motion A level FM2 AS level Mech b
Momentum & restitution AS level FM1 AS AS level Mech a
Definitions
2